where \(\sigma _i\) , \(\tau _i\) , \(\mathbf {s}_i\)  (\(i=x,y,z\) ) are, respectively,
the Pauli matrices for sublattice isospin, layer isospin and real
spin. Here, \(\xi =\pm 1\)  indicate two opposite valleys \(K\) , \(K^{\prime }\)  of the
hexagonal Brillouin zone, \(p_{x(y)}\)  is the momentum component,
\(\upsilon _0=3at/2\)  is the Fermi velocity in graphene (\(a\) , \(t\)  are
the bond length and nearest-neighbor hopping, respectively),
and \(\gamma \) , \(m_z\)  and \(\lambda _U\)  denote the strengths of the
interactions. The first and second terms are, respectively, the massless Dirac term
and the interlayer nearest-neighbor coupling
(usually sufficient to capture the band features
[1]}, [2]}). The third term indicates the LAF field induced by short-range magnetic proximity from two FM layers [3]}, [4]}. The last term represents the vertical bias.
